Band structure engineering through orbital

interaction for enhanced thermoelectric power

factor

Hong Zhu, Wenhao Sun, Rickard Armiento, Predrag Lazić and Gerbrand Ceder

Linköping University Post Print

N.B.: When citing this work, cite the original article.

Original Publication:

Hong Zhu, Wenhao Sun, Rickard Armiento, Predrag Lazić and Gerbrand Ceder, Band

structure engineering through orbital interaction for enhanced thermoelectric power factor,

2014, Applied Physics Letters, (104), 082107-1-082107-5.

http://dx.doi.org/10.1063/1.4866861

Copyright: American Institute of Physics (AIP)

http://www.aip.org/

Postprint available at: Linköping University Electronic Press

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-105274

Band structure engineering through orbital interaction for enhanced thermoelectricpower factorHong Zhu, Wenhao Sun, Rickard Armiento, Predrag Lazic, and Gerbrand Ceder

Citation: Applied Physics Letters 104, 082107 (2014); doi: 10.1063/1.4866861 View online: http://dx.doi.org/10.1063/1.4866861 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/104/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Camel-back band-induced power factor enhancement of thermoelectric lead-tellurium from Boltzmann transportcalculations Appl. Phys. Lett. 104, 132106 (2014); 10.1063/1.4870494 Energy bands, conductance, and thermoelectric power for ballistic electrons in a nanowire with spin-orbitinteraction J. Appl. Phys. 108, 093704 (2010); 10.1063/1.3493113 Minimum length scales for enhancement of the power factor in thermoelectric nanostructures J. Appl. Phys. 107, 074304 (2010); 10.1063/1.3359659 Thermoelectric power and structural properties in two-phase Sn/SnTe alloys J. Appl. Phys. 105, 023713 (2009); 10.1063/1.3068463 Thermoelectric power factor in superlattice systems Appl. Phys. Lett. 77, 705 (2000); 10.1063/1.127092

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.236.83.168 On: Tue, 22 Apr 2014 10:44:41

Band structure engineering through orbital interaction for enhancedthermoelectric power factor

Hong Zhu,1 Wenhao Sun,1 Rickard Armiento,2 Predrag Lazic,3 and Gerbrand Ceder1

1Department of Materials Science and Engineering, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA2Department of Physics, Chemistry and Biology (IFM), Link€oping University, SE-58183 Link€oping, Sweden3Theoretical Physics Division, Rudjer Boskovic Institute, Bijenicka Cesta 54, Zagreb, Croatia

(Received 12 January 2014; accepted 10 February 2014; published online 26 February 2014)

Band structure engineering for specific electronic or optical properties is essential for the further

development of many important technologies including thermoelectrics, optoelectronics, and

microelectronics. In this work, we report orbital interaction as a powerful tool to finetune the band

structure and the transport properties of charge carriers in bulk crystalline semiconductors. The

proposed mechanism of orbital interaction on band structure is demonstrated for IV-VI

thermoelectric semiconductors. For IV-VI materials, we find that the convergence of multiple

carrier pockets not only displays a strong correlation with the s-p and spin-orbit coupling but also

coincides with the enhancement of power factor. Our results suggest a useful path to engineer the

band structure and an enticing solid-solution design principle to enhance thermoelectric

performance. VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4866861]

Thermoelectrics are a potentially important technology

for the conversion of heat flows to electrical work and vice-

versa.1 The thermoelectric figure of merit, zT, can be

increased by minimizing the thermal conductivity (j) and/or

maximizing the power factor (rS2), where r and S are the

electrical conductivity and Seebeck coefficient.2 Over the

last two decades, significant progress has been made to

reduce the lattice component of thermal conductivity (jL)

via solid-solution alloying and nanostructuring.3–6 With jL

approaching its amorphous limit,7 further enhancement of zTmay have to come from improvement of rS2.8 The optimal

rS2 is dependent on lðm�=meÞ3=2, where l is the carrier mo-

bility and m* and me are the density-of-states effective mass

and the electron rest mass.9,10 In a single parabolic band

model, this strategy requires conflicting band curvatures—

flat bands will result in large m*, but dramatically reduce l.

Recently, a strategy to address this dilemma has appeared,

focusing on increasing the number of degenerate carrier

pockets at the Fermi energy, since it maximizes m* without

sacrificing l of each band.10 This can happen when multiple

bands have band extrema around the same energy (orbital

degeneracy) or when multiple carrier pockets of one band in

the Brillouin zone are degenerate (valley degeneracy).11

The success of increasing valley degeneracy for high

thermoelectric performance has been demonstrated in the

premier thermoelectric material, lead telluride (PbTe). PbTe

is a small band gap semiconductor with two band extrema

for both conduction and valence bands—the band extrema at

the L point (with a multiplicity of 4 in the whole Brillouin

zone) and an additional, heavier pocket along the R line

(with a multiplicity of 12) (see Fig. 1). By alloying p-type

PbTe with appropriate amounts of PbSe, a high valley degen-

eracy of 16 is obtained for the valence band due to the

convergence of the L and R pockets. This leads to a simulta-

neous enhancement of S and r, as well as a high zT value of

1.8, compared to 0.8 for pure PbTe.10 Another example of

such band engineering is the doping of Mg2Si with Sn to

increase the orbital degeneracy by aligning the two

low-lying conduction bands, leading to an exceptionally

large zT value of 1.3.12 Despite the experimental success of

such isostructural solid-solution alloying, the fundamental

mechanisms that allow manipulation of band structures via

handles, such as chemical composition, are presently not

well understood. Because of this, it is unclear a prioriwhether or not a given thermoelectric material can be

improved, and even if it can be, what design parameters can

be used for efficient band convergence.

In this work, we report a strong correlation between the

orbital interaction and particular band structure features such

as band degeneracy. We find that the strength of the orbital

interaction can be tuned by manipulating the interatomic

distance or by chemically varying the relative positions

of atomic orbitals through composition modification.

Specifically, for IV-VI thermoelectric semiconductors, i.e.,

lead/tin/germanium chalcogenides, we find that the conver-

gence of carrier pockets is dependent on the strength of the

s-p orbital interaction as well as the spin-orbit coupling

(SOC). Our transport calculations also demonstrate that the

enhancement of power factor coincides with such a band

convergence. Although this work investigates the IV-VI

semiconductors as a prototype for explaining the proposed

mechanism of orbital interaction on band structure, our con-

clusions are applicable to other systems (e.g., Sb2Te3 group

materials with s-p mixing and ZnO group materials with p-dmixing). Our results suggest a path to engineer the band

structure, i.e., towards high band degeneracy for enhanced

thermoelectric properties.

Density functional theory (DFT) calculations were per-

formed with the Perdew-Burke-Ernzerhof (PBE)13 general-

ized gradient approximation (GGA) and projector

augmented wave (PAW)14 pseudopotentials, as implemented

in Vienna ab initio simulation package (VASP).15 For the

geometry relaxation of the primitive unit cell of PbTe group

materials in the rock-salt (NaCl) structure, we considered

0003-6951/2014/104(8)/082107/5/$30.00 VC 2014 AIP Publishing LLC104, 082107-1

APPLIED PHYSICS LETTERS 104, 082107 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.236.83.168 On: Tue, 22 Apr 2014 10:44:41

spin-orbit coupling and applied a k-point mesh of

12� 12� 12. The calculated equilibrium lattice parameters

are in good agreement with available experiments as shown

in Table I. The electronic structure used to determine the

transport properties were obtained from calculations with

much denser k-point meshes (23� 23� 23). Using the

Boltzmann transport equation and constant relaxation-time

approximation as implemented in the BOLTZTRAP

code,16,17 we calculated the maximum power factor relative

to the relaxation time, S2r/s. Since the corresponding Fermi

levels for optimal S2r/s of holes and electrons are deep

within the valence and conduction bands, respectively, fur-

ther correction to Eg using a scissor operator is not consid-

ered in our transport calculations.

Using PbTe as an example, we now discuss the elec-

tronic structure of IV-VI semiconductors. Fig. 1(a) schemati-

cally draws the molecular orbital (MO) diagram of a PbTe

molecule. Due to the s-p coupling effect, the r bond has pre-

dominantly Te-p and Pb-s character, while r* mainly con-

sists of Pb-p and Te-s. With a stronger s-p coupling, r and

r* bonds both move upward. We also note that due to the

larger energy difference between Pb-p and Te-s orbitals

compared to Te-p and Pb-s, the s-p mixing strength of the r*

bond is smaller than that of the r bond. Fig. 1(b) shows the

relevant part of the band structure of PbTe at the optimized

lattice parameter with the inclusion of SOC, which is neces-

sary to represent the band splitting of this group of materi-

als.19 The band gap of PbTe is direct and occurs at the L

point. The top of the valence band and the bottom of the con-

duction band correspond to the r and r* bonds in the MO

picture.19,20 The valence band maxima at L and R are mainly

composed of Te-px,y,zþ Pb-s and Te-px,yþ Pb-s, as repre-

sented by Lanion and R anion. The two conduction band min-

ima (CBM) (Lcation and Rcation) have similar characters,

except that the identity of Pb and Te are interchanged (see

supplementary material in Ref. 21). A high valley degener-

acy of 16 in the Brillouin zone can be obtained by aligning

the L and R carrier pockets of the conduction/valence band.

Such an alignment can be characterized by the energy differ-

ence between the L and R valleys, namely, (L–R)C or

(L–R)V. In fact, the number of conducting carrier pockets

can be further increased to 20 with an enhanced orbital

degeneracy by converging two bands at the L point. Lanion

and Lcation have symmetries of Lþ6 and L�6 .19 With the inclu-

sion of SOC, the p (or p*) bond in the MO diagram splits

into two states at the L point in the band structure diagram,

namely, Lþ4;5 and Lþ6 (or L�4;5 and L�6 ) (see Fig. 2). The orbital

FIG. 1. Schematic of (a) MO diagram of a PbTe molecule and (b) band

structure of solid state NaCl-type PbTe at the optimized lattice parameters,

with the high lying valence band and low lying conduction band projected

onto the Pb and Te atomic orbital makeup—the curve widths indicate the

relative strengths of each contribution. The valence band maxima, namely,

Ranion and Lanion, are mainly composed of Te-px,yþPb-s and Te-px,y,z

þPb-s, respectively. The two conduction band minima, Rcation and Lcation,

have similar characters, except that the identity of Pb and Te are inter-

changed. The energy difference between the two highest valence bands (or

between the two lowest conduction bands) at the L point of the Brillouin

zone is represented as DLV (or DLC).

TABLE I. Computed lattice parameter and band gap (Eg) using PAW

method with SOC compared with the available experiments.18 Eg is defined

as the energy difference between the cation-p orbital dominated states and

the anion-p dominated states around the Fermi-level. A negative Eg indicates

inversion of these around the gap.

Eg (eV)

Lattice parameter (A) Expt

System PAW-SOC Expt PAW-SOC 4.2 K 300 K

PbTe 6.556 6.462 0.09 0.19 0.31

PbSe 6.182 6.124 �0.05 0.15 0.28

PbS 5.981 5.936 0.11 0.29 0.41

SnTe 6.40 6.327 �0.12 … �0.18

SnSe 6.041 … �0.24 … …

SnS 5.829 5.8 �0.18 … …

GeTe 6.012 5.985 0.23 … …

GeSe 5.655 … �0.06 … …

GeS 5.428 … �0.07 … …

FIG. 2. Band structure variation upon hydrostatic strain: (a) eigenvalues of

Ranion, Lanion, Rcation and Lcation states, and (b) eigenvalues of three highest

valence states and three lowest conduction states at the L point. Valley con-

vergence [(L-R)V and (L-R)C], orbital convergence at L point (DLV and

DLC), and Eg are indicated in the figure.

082107-2 Zhu et al. Appl. Phys. Lett. 104, 082107 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.236.83.168 On: Tue, 22 Apr 2014 10:44:41

degeneracy can be characterized by the energy difference

between the two highest valence bands (or between the two

lowest conduction bands) at the L point, which is represented

as DLV (or DLC) in Fig. 1.

One approach to tune the s-p mixing is through strain,

which can result from an externally applied stress, thermal

expansion, alloying/doping, or lattice mismatch in hetero-

structures. Here, we use PbTe as an example to demonstrate

how hydrostatic strain affects the band structure by tuning

the orbital interaction strength. In general, the s-p mixing

strengths of the four band extrema in PbTe follow the trend

of Lanion>Ranion>Lcation>Rcation, in light of different sym-

metries and s-p energy splits. But due to the fact that the two

low-lying cation levels at the L point have the same symme-

try of L�6 , they tend to repel each other (see Fig. 2). This

repulsion not only leads to a much larger (L–R)C than

(L–R)V but also restricts the rise of Lcation with the increase

of s-p coupling upon compression. The dependence of the

four band extrema on strain thus follows the trend of

Lanion>Ranion>Rcation>Lcation (see Fig. 2(a)). Since com-

pressive strain can raise Lanion above Lcation, the valley con-

vergence can be described as

ðL� RÞV ¼ minðLanion;LcationÞ � Ranion; (1)

ðL� RÞC ¼ Rcation �maxðLanion;LcationÞ: (2)

As shown in Fig. 2(a), (L–R)V and (L–R)C of PbTe first

increase with compressive strain. Once the critical strain

where Lanion and Lcation cross is approached, they start to

decrease with further compression.

Fig. 2(b) illustrates the orbital convergence of PbTe at

the L point as a function of strain. With the increase of Te-pand Pb-s orbital coupling, the Lanion (the higher Lþ6 ) state

shifts away from the lower anion L state (Lþ4;5), which results

in a larger DLV. In other words, extensive strain appears to

increase the orbital degeneracy of holes. On the other hand,

the symmetry-induced repulsion prevents Lcation (the lower

L�6 ) state from collapsing to the next L�6 state. Applying a

compressive strain to invert the higher Lþ6 and the lower L�6states avoids such a repulsion and leads to a smaller DLC.

Thus, a high orbital degeneracy of the valence band can be

achieved upon extensive strain, while for the conduction

band it can be enhanced mainly upon compression.

Previously, we have demonstrated that strain could be

applied to tune the valley and orbital degeneracies. Next, we

discuss the feasibility of chemically shifting the relative

position of the s and p atomic orbitals to adjust their cou-

pling strengths. Table I shows the computed Eg for IV-VI

semiconductors at their optimized lattice parameter. Eg is

defined as the difference in energy of the states dominated

by the cation p and anion p orbitals around the Fermi level.

For PbTe at relaxed lattice parameter, this is the CBM and

VBM, respectively. Relative to PbTe, some of the com-

pounds (e.g., SnTe) have opposite orbital characters at the

band extrema,22 which we indicate by a negative Eg. As

summarized in Fig. 3, the band degeneracies of IV-VI mate-

rials display a similar dependence on strain. More interest-

ingly, we find that these band degeneracies have a strong

correlation with the relative position of s and p orbitals of

the two constituent elements (see Figs. 3 and 4). For the va-

lence band, we find that a larger energy split between anion

p and cation s orbitals (Eanion�p � Ecation�s) gives a higher

band degeneracy (or smaller (L–R)V and DLV). An example

of (L–R)V as a function of Eanion�p � Ecation�s for IV-VI

semiconductors at a fixed lattice parameter of 6.1 A is pro-

vided in Fig. 4(b). For the conduction band, both SOC and

s-p coupling determine the band degeneracy. Different from

the valence band, a high conduction band degeneracy can be

FIG. 3. Valley convergence (L–R) and

orbital convergence (DL) as a function

of strain for IV-VI thermoelectric

semiconductors, which are shown in

black and red. The solid and dashed

lines represent the data for valence and

conduction bands, respectively. The

compressive strain is shown upto the

value when Eg is zero due to the over-

lapping of the conduction and valence

bands. See supplementary material for

calculated Eg as a function of strain.21

082107-3 Zhu et al. Appl. Phys. Lett. 104, 082107 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.236.83.168 On: Tue, 22 Apr 2014 10:44:41

obtained with a small energy split between cation p and

anion s orbital (Ecation�p � Eanion�s) (see Fig. 4(c) for the

example of the valley convergence of conduction bands).

Moreover, when we go from Pb ! Ge, the SOC effect

decreases significantly, which leads to a reduced repulsion

of the two L�6 states and a better convergence of conduction

bands. By investigating the IV-VI group materials under dif-

ferent strains, we have noticed that: (1) the valence band

extrema can be converged by increasing Eanion�p � Ecation�s;

(2) a high conduction band degeneracy can be achieved

with a reduced Ecation�p � Eanion�s and/or a smaller SOC

of cations.

We next investigate the optimal power factor (S2r/s).

Here, we focus on the results of lead chalcogenides (see

Fig. 5). Tin and germanium chalcogenides present similar

results and hence are not shown here. When we go from

PbTe to PbS, the optimal S2r/s decreases for both holes and

electrons, which can be attributed to the better band conver-

gence of PbTe. For holes, we find that the optimal power fac-

tor is strongly correlated to the band convergence. This

correlation is most notorious in PbSe and PbS, while in the

case of PbTe, the L and R valleys of valence bands are close

enough to be effectively converged at high temperatures, due

to the broadening of the Fermi distribution.10 In light of the

strong SOC effect of Pb, a large power factor of electrons

can be only achieved upon a compressive strain when there

is no repulsion between CBM and the higher L�6 state. On

the other hand, Ge- and Sn-based chalcogenides with smaller

SOC in the cations present higher n-type S2r/s, due to the

enhanced conduction band degeneracy.

We are thus led to conclude that s-p orbital interaction

and spin-orbit coupling can be used to engineer the band

structure and the transport properties of IV-VI semiconduc-

tors. Such a tunability of band structure with orbital interac-

tion should maintain its generality in other materials with

similar interactions, but the quantitative correlation may

need further investigation in each group of materials. For

example, the band gap of II-VI semiconductors varies with

the p-d repulsion of the VBM. Compared to ZnTe and CdTe,

HgTe with a smaller energy split between the cation d and

anion p orbitals shows a stronger orbital repulsion and hence

a smaller band gap.23 Similar to alloying and quantum con-

finement,24,25 orbital interaction may serve as a powerful

tool to fine-tune the electronic structure of a material in dif-

ferent application areas such as thermoelectrics, optoelec-

tronics, and microelectronics.

FIG. 4. (a) Atomic orbital energy (eV) of anions (Te, Se, S) and cations (Pb, Sn, Ge), (b) valley convergence of valence band, (L–R)V, as a function of energy

split between anion p and cation s orbitals (Eanion�p � Ecation�s), and (c) valley convergence of conduction band, (L–R)C, as a function of energy split between

cation p and anion s orbitals (Ecation�p � Eanion�s). The thin and thick bars represent the s and p orbitals, respectively. The lattice parameter is chosen to be

6.1 A for (b) and (c) to identify the correlation between band convergence and energy split between atomic orbitals.

FIG. 5. The optimal power factor

(S2r/s in the unit of W/m K2 s), of lead

chalcogenides for (a) holes and (b)

electrons as a function of strain (%)

and temperature (K). The color scale is

indicated in the figure.

082107-4 Zhu et al. Appl. Phys. Lett. 104, 082107 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.236.83.168 On: Tue, 22 Apr 2014 10:44:41

The results and insights demonstrated how alloying can

be used to modify the band structure in a well determined

way with the objective of enhancing the band degeneracy

and power factor of semiconductors. We note from Fig. 3

that both compressive and extensive strains can enhance the

valence band degeneracy. Thus, one may want to alloy a

p-type IV-VI semiconductor so as to lead to a better band

convergence. For example, the valence band degeneracy of

PbSe can be enhanced through PbTe alloying. However,

sometimes we may have an “over-converged” band structure

with L-R< 0. In that case, the reverse action should be taken

to bring L-R back to zero. It has been reported that (L-R)V of

PbTe decreases as temperature increases due to thermal

expansion. At 850 K, PbTe displays a negative (L-R)V.10

Alloying PbTe with 15% PbSe realigns the L and R carrier

pockets of valence bands at such a high temperature and

leads to a large zT of 1.8. Such a recovery of (L-R)V from

negative to zero could be understood as follows: first, PbSe

has a larger (L-R)V compared to PbTe; second, the inclusion

of PbSe with smaller lattice parameter introduces a negative

strain into PbTe1�xSex, which compensates its thermal

expansion. For electrons, the introduction of compressive

strains and cations with small SOC appears to be efficient at

converging the conduction band extrema. Thus, a preferred

alloy phase for n-type IV-VI materials shall possess smaller

lattice parameter or SOC in cations. For example,

Pb1�xGexTe is likely to be good n-type thermoelectric mate-

rials. In fact, it has been reported that 5% GeTe inclusion

into PbTe was able to increase the zT from 0.8 to �1.3.26

In summary, we have explained recent experimental

advances in increased power factor for thermoelectric mate-

rials in terms of orbital interactions. Our results suggest a

general strategy for engineering band convergence by strain

and composition modification. For IV-VI thermoelectric

materials, we found that holes and electrons behave differ-

ently with these modifications due to the repulsion within

the conduction bands. For holes, one can either apply

appropriate compressive/extensive strain or split the anion

p and cation s orbitals to converge the valence bands for

enhanced power factors. For electrons, a higher band

degeneracy may be achieved with small-SOC cations or

upon compressive strains which lift Lanion over Lcation and

hence eliminate the symmetry-induced repulsion within the

conduction bands. Finally, we have proposed an alloying

strategy to optimize the available IV-VI thermoelectric

materials, which has been validated by prior experiments in

some cases.

This work was supported by the Office of Naval Research

under Contract No. N00014-11-1-0212. R.A. acknowledges

support from the Swedish Research Council (VR), Grant No.

621-2011-4249 and the Linnaeus Environment at Link€oping

on Nanoscale Functional Materials (LiLi-NFM) funded by

VR. We would also like to thank Professor Jeff Snyder and

Vidvuds Ozolins for helpful discussions.

1L. E. Bell, Science 321, 1457 (2008).2J. R. Sootsman, D. Y. Chung, and M. G. Kanatzidis, Angew. Chem., Int.

Ed. 48, 8616 (2009).3B. Poudel, Q. Hao, Y. Ma, Y. Lan, A. Minnich, B. Yu, X. Yan, D. Wang,

A. Muto, D. Vashaee et al., Science 320, 634 (2008).4J. Tang, H. Wang, D. H. Lee, M. Fardy, Z. Huo, T. Russell, and P. Yang,

Nano Lett. 10, 4279 (2010).5J. Yu, S. Mitrovic, D. Tham, J. Varghese, and J. R. Heath, Nat.

Nanotechnol. 5, 718 (2010).6K. Biswas, J. He, I. D. Blum, C. Wu, T. P. Hogan, D. N. Seidman, V. P.

Dravid, and M. G. Kanatzidis, Nature 489, 414 (2012).7C. J. Vineis, A. Shakouri, A. Majumdar, and M. G. Kanatzidis, Adv.

Mater. 22, 3970 (2010).8J. P. Heremans, V. Jovovic, E. S. Toberer, A. Saramat, K. Kurosaki, A.

Charoenphakdee, S. Yamanaka, and G. J. Snyder, Science 321, 554

(2008).9H. J. Goldsmid, J. Electron. Mater. 42, 1482 (2013).

10Y. Pei, X. Shi, A. LaLonde, H. Wang, L. Chen, and G. J. Snyder, Nature

473, 66 (2011).11Y. Pei, H. Wang, and G. J. Snyder, Adv. Mater. 24, 6125 (2012).12W. Liu, X. Tan, K. Yin, H. Liu, X. Tang, J. Shi, Q. Zhang, and C. Uher,

Phys. Rev. Lett. 108, 166601 (2012).13J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865

(1996); 78, 1396 (1997).14P. E. Bl€ochl, Phys. Rev. B 50, 17953 (1994).15G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996).16G. K. H. Madsen, J. Am. Chem. Soc. 128, 12140 (2006).17G. K. H. Madsen and D. J. Singh, Comput. Phys. Commun. 175, 67 (2006).18Semiconductors: Group IV Elements, IV-IV and III-IV Compounds,

Landolt-B€ornstein, New Series, Group III, Vol. 41, Part A, edited by O.

Madelung, U. Rossler, and M. Schulz (Springer-Verlag, Berlin, 2005);

B. F. Bilenkii, A. G. Mikolaichuk, and D. M. Freik, Phys. Status Solidi 28,

K5 (1968); T. Chattopadhyay, J. X. Boucherle, and H. G. vonSchnering,

J. Phys. C: Solid State Phys. 20, 1431 (1987).19S.-H. Wei and A. Zunger, Phys. Rev. B 55, 13605 (1997).20H. Peng, J. Song, M. G. Kanatzidis, and A. J. Freeman, Phys. Rev. B 84,

125207 (2011).21See supplementary material at http://dx.doi.org/10.1063/1.4866861 for or-

bital projected band structure of PbTe at relaxed lattice parameter and the

computed Eg for lead, tin, and germanium chalcogenides as a function of

hydrostatic strain.22T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, and L. Fu, Nat. Commun.

3, 982 (2012).23S.-H. Wei and A. Zunger, Phys. Rev. B 37, 8958 (1988).24I. Vurgaftman and J. R. Meyer, J. Appl. Phys. 94, 3675 (2003).25L. D. Hicks and M. S. Dresselhaus, Phys. Rev. B 47, 12727 (1993).26I. Kudman, Metal. Trans. 2, 163 (1971).

082107-5 Zhu et al. Appl. Phys. Lett. 104, 082107 (2014)

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.236.83.168 On: Tue, 22 Apr 2014 10:44:41